Approximation and hardness results for the Max k-Uncut problem

Abstract

In the study of the homophily law of large scale complex networks, we get a combinatorial optimization problem which we call the Maxk-Uncut problem. Given an n-vertex undirected graph G=(V,E) with nonnegative weights {we|e∈E} defined on edges, and a positive integer k, the Maxk-Uncut problem asks to find a partition {V1,V2,⋯,Vk} of V such that the total weight of edges that are not cut is maximized. Intuitively, an edge that is not cut connects two vertices with the same or similar attributes since they are in the same part of the partition. Interestingly, the Maxk-Uncut problem is just the complement of the classic Mink-Cut problem. For Maxk-Uncut, we present a randomized (1−kn)2-approximation algorithm, a greedy (1−2(k−1)n)-approximation algorithm, and an Ω(12α)-approximation algorithm by reducing it to Densestk-Subgraph, where α is the approximation ratio of the Densestk-Subgraph problem. More importantly, we show that Maxk-Uncut and Densestk-Subgraph are in fact equivalent in approximability up to a factor of 2. We also prove an approximation hardness result for Maxk-Uncut under the assumption P≠NP.